Linear Models and Effect Magnitudes for Research,Clinical and 线性模型和影响大小和临床研究

Ifyouareviewingthisslideshowwithinabrowserwindow,selectFile/Saveas…fromthetoolbarandsavetheslideshowtoyourcomputer,thenopenitdirectlyinPowerPoint.Whenyouopenthefile,usethefull-screenviewtoseetheinformationoneachslidebuildsequentially.Forfull-screenview,clickonthisiconontheloweredgeofthePowerpointwindow.Togoforwards,left-clickorhitthespacebar,PdDnorkey.Togobackwards,hitthePgUporkey.Toexitfromfull-screenview,hittheEsc(escape)key.Fitness

0100Chances

selected

(%)abdc

strengthpostpreAgeActivity

r

=

0.57LinearModelsandEffectMagnitudesforResearch,ClinicalandPracticalApplicationsWillGHopkins

AUTUniversity,Auckland,NZSportscience14,49-57,2021

(/2021/wghlinmod)ImportanceofEffectMagnitudesGettingEffectsfromModelsLinearmodels;adjustingforcovariates;interactions;polynomialsEffectsforacontinuousdependentDifferencebetweenmeans;“slope〞;correlationGenerallinearmodels:ttests;multiplelinearregression;ANOVA…Uniformityoferror;logtransformation;within-subjectandmixedmodelsEffectsforanominalorcountdependentRiskdifference;risk,odds,hazardandcountratiosGeneralizedlinearmodels:Poisson,logistic,log-hazardProportional-hazardsregressionBackground:TheRiseofMagnitudeofEffectsResearchisallabouttheeffectofsomethingonsomethingelse.Thesomethingsarevariables,suchasmeasuresofphysicalactivity,health,training,performance.Aneffectisarelationshipbetweenthevaluesofthevariables,forexamplebetweenphysicalactivityandhealth.Wethinkofaneffectascausal:moreactivemorehealthy.Butitmaybeonlyanassociation:moreactivemorehealthy.Effectsprovideuswithevidenceforchangingourlives.Themagnitudeofaneffectisimportant.Inclinicalorpracticalsettings:couldtheeffectbeharmfulorbeneficial?Isthebenefitlikelytobesmall,moderate,large…?Inresearchsettings:Effectmagnitudedeterminessamplesize.Meta-analysisisallaboutaveragingmagnitudesofstudy-effects.SovariousresearchorganizationsnowemphasizemagnitudeGettingEffectsfromModelsAneffectarisesfromadependentvariableandoneormorepredictor(independent)variables.Therelationshipbetweenthevaluesofthevariablesisexpressedasanequationormodel.Exampleofonepredictor:Strength=a+b*AgeThishasthesameformastheequationofaline,Y=a+b*X,hencethetermlinearmodel.Themodelisusedasifitmeans:Strengtha+b*Age.IfAgeisinyears,themodelimpliesthatoldersubjectsarestronger.Themagnitudecomesfromthe“b〞coefficientorparameter.Realdatawon’tfitthismodelexactly,sowhat’sthepoint?Well,itmightfitquitewellforchildrenoroldfolks,andifso…Wecanpredicttheaveragestrengthforagivenage.Andwecanassesshowfaroffthetrendagivenindividualfalls.Exampleoftwopredictors:Strength=a+b*Age+c*SizeAdditionalpredictorsaresometimesknownascovariates.ThismodelimpliesthatAgeandSizehaveeffectsonstrength.It’sstillcalledalinearmodel(butit’saplanein3-D).Linearmodelshaveanincredibleproperty:theyallowustoworkoutthe“pure〞effectofeachpredictor.BypurehereImeantheeffectofAgeonStrengthforsubjectsofanygivenSize.Thatis,whatistheeffectofAgeifSizeisheldconstant?Thatis,yeah,kidsgetstrongerastheygetolder,butisitjustbecausethey’rebigger,ordoessomethingelsehappenwithage?Thesomethingelseisgivenbythe“b〞:ifyouholdSizeconstantandchangeAgebyoneyear,Strengthincreasesbyexactly“b〞.WealsorefertotheeffectofAgeonStrengthadjustedforSize,controlledforSize,or(recently)conditionedonSize.Likewise,“c〞istheeffectofoneunitincreaseinSizeforsubjectsofanygivenAge.Withkids,inclusionofSizewouldreducetheeffectofAge.Tothatextent,SizeisamechanismormediatorofAge.Butsometimesacovariateisaconfounderratherthanamediator.Example:PhysicalActivity(predictor)hasastrongrelationshipwithHealth(dependent)inelderlyadults.Ageisaconfounderoftherelationship,becauseAgecausesbadhealthandinactivity.Again,includingpotentialconfoundersascovariatesproducesthepureeffectofapredictor.Thinkcarefullywheninterpretingtheeffectofincludingacovariate:isthecovariateamechanismoraconfounder?IfyouareconcernedthattheeffectofAgemightdifferforsubjectsofdifferentSize,youcanaddaninteraction…Exampleofaninteraction:

Strength=a+b*Age+c*Size+d*Age*SizeThismodelimpliesthattheeffectofAgeonStrengthchangeswithSizeinsomesimpleproportionalmanner(andviceversa).It’sstillknownasalinearmodel.YoustillusethismodeltoadjusttheeffectofAgefortheeffectofSize,buttheadjustedeffectchangeswithdifferentvaluesofSize.Anotherexampleofaninteraction:

Strength=a+b*Age+c*Age*Age=a+b*Age+c*Age2ByinteractingAgewithitself,yougetanon-lineareffectofAge,hereaquadratic.Ifcturnsouttobenegative,thismodelimpliesstrengthrisestoamaximum,thencomesdownagainforoldersubjects.Tomodelsomethingfallingtoaminimum,cwouldbepositive.Tomodelmorecomplexcurvature,addd*Age3,e*Age4…Thesearecubics,quartics…,butit’sraretogoaboveaquadratic.Thesemodelsarealsoknownaspolynomials.Theyareallcalledlinearmodels,eventhoughtheymodelcurves.Usethecoefficientstogetdifferencesbetweenchosenvaluesofthepredictor,andvaluesofpredictoranddependentatmaxormin.Complexcurvatureneedsnon-linearmodeling(seelater)orlinearmodelingwiththepredictorconvertedtoanominalvariable…Group,factor,classificationornominalvariablesaspredictors:WehavebeentreatingAgeasanumberofyears,butwecouldinsteaduseAgeGroup,withseverallevels;e.g.,child,adult,elderly.Statspackagesturneachlevelintoadummyvariablewithvaluesof0and1,thentreateachasanumericvariable.Example:Strength=a+b*AgeGroupistreatedas

Strength=a+b1*Child+b2*Adult+b3*Elderly,

whereChild=1forchildrenand0otherwise,Adult=1foradultsand0otherwise,andElderly=1foroldfolkand0otherwise.Themodelestimatesthemeanvalueofthedependentforeachlevelofthepredictor:meanstrengthofchildren=a+b1.Andthedifferenceinstrengthofadultsandchildrenisb2–b1.Youdon’tusuallyhavetoknowaboutcodingofdummies,butyoudowhenusingSPSSforsomemixedmodelsandcontrolledtrials.Dummyvariablescanalsobeveryusefulforadvancedmodeling.Forsimpleanalysesofdifferencesbetweengroupmeanswitht‑tests,youdon’thavetothinkaboutmodelsatall!LinearmodelsforcontrolledtrialsForastudyofstrengthtrainingwithoutacontrolgroup:

Strength=a+b*Trial,whereTrialhasvaluespre,postorwhatever.b*Trialisreallyb1*Pre+b2*Post,withPre=1or0andPost=1or0.Theeffectoftrainingonmeanstrengthisgivenbyb2–b1.Forastudywithacontrolgroup:

Strength=a+b*Group*Trial,whereGrouphasvaluesexpt,cont.b*Group*Trialisreally

b1*ContPre

+

b2*ContPost

+

b3*ExptPre

+

b4*ExptPost.Thechangesinthegroupsaregivenbyb2–b1andb4–b3.Theneteffectoftrainingisgivenby(b4–b3)–(b2–b1).Statspackagesalsoallowyoutospecifythismodel:

Strength=a+b*Group+c*Trial+d*Group*Trial.GroupandTrialaloneareknownasmaineffects.Thismodelisreallythesameastheinteraction-onlymodel.Itdoesalloweasyestimationofoverallmeandifferencesbetweengroupsandmeanchangespretopost,buttheseareuseless.Oryoucanmodelchangescoresbetweenpairsoftrials.Example:Strength=a+b*Group*Trial,wherebhasfourvalues,

isequivalentto

StrengthChange=a+b*Group,wherebhasjusttwovalues(exptandcont)andStrengthChangeisthepost-prechangescores.Youcanincludesubjectcharacteristicsascovariatestoestimatethewaytheymodifytheeffectofthetreatment.Suchmodifiersormoderatorsaccountforindividualresponsestothetreatment.Apopularmodifieristhebaseline(pre)scoreofthedependent:

StrengthChange=a+b*Group+c*Group*StrengthPre.Herethetwovaluesofcestimatethemodifyingeffectofbaselinestrengthonthechangeinstrengthinthetwogroups.Andc2–c1isthenetmodifyingeffectofbaselineonthechange.Bonus:abaselinecovariateimprovesprecisionofestimationwhenthedependentvariableisnoisy.Modelingofchangescoreswithacovariateisbuiltintothecontrolled-trialspreadsheetsatSportscience.Youcanincludethechangescoreofanothervariableasacovariatetoestimateitsroleasamediator(i.e.,mechanism)ofthetreatment.

Example:StrengthChange=a+b*Group+d*MediatorChange.drepresentshowwellthemediatorexplainsthechangeinstrength.b2–b1istheeffectofthetreatmentwhenMediatorChange=0;

thatis,theeffectofthetreatmentnotmediatedbythemediator.

Linearvsnon-linearmodelsAnydependentequaltoasumofpredictorsand/ortheirproductsisalinearmodel.Anythingelseisnon-linear,e.g.,anexponentialeffectofAge,tomodelstrengthreachingaplateauratherthanamaximum.Almostallstatisticalanalysesarebasedonlinearmodels.Andtheycanbeusedtoadjustforothereffects,includingestimationofindividualresponsesandmechanisms.Non-linearproceduresareavailablebutaremoredifficulttouse.SpecificLinearModels,EffectsandThresholdMagnitudesThesedependonthefourkinds(ortypes)ofvariable.Continuous(numberswithdecimals):mass,distance,time,current;measuresderivedtherefrom,suchasforce,concentration,volts.Counts:suchasnumberofinjuriesinaseason.Ordinal:valuesarelevelswithasenseofrankorder,suchasa

4-ptLikertscaleforinjuryseverity(none,mild,moderate,severe).Nominal:valuesarelevelsrepresentingnames,suchas

injured(no,yes),andtypeofsport(baseball,football,hockey).Aspredictors,thefirstthreecanbesimplifiedtonumeric.Ifapolynomialisinappropriate,parseinto2-5levelsofanominal.Example:AgebecomesAgeGroup(5-14,15-29,30-59,60-79,>79).Valuescanalsobeparsedintoequalquantiles(e.g.,quintiles).IfanordinalpredictorsuchasaLikertscalehasonly2-4levels,orifthevaluesarestackedatoneendofthescale,analyzethevaluesaslevelsofanominalvariable.Asdependents,eachtypeofvariableneedsadifferentapproach.Continuousvariables(e.g.,time)andordinalswithenoughlevels(e.g.,7-ptLikertresponsesortheirsums)needvariousformsofgenerallinearmodelandgeneralmixedlinearmodel.ThesemodelsareunifiedbytheassumptionthattheoutcomestatistichasaTsamplingdistribution.Generalizedlinearmodelsandgeneralizedmixed

linearmodelsareusedwithbinarynominalvariablescodedintovaluesof0or1(e.g.,injuredornot,rugbyornot)andwithcountscodedasaninteger(e.g.,numberofinjuries).Thesemodelstakeintoaccountthespecialdistributionsofthedependentvariable:binomial(for0and1)andPoisson(forcounts).Thegenerallinearmodelisoneofthegeneralizedlinearmodels.Ordinalvariableswithonlyafewlevelsandnominalswithseverallevelseitherneedspecificformsofgeneralizedlinearmodelorthelevelscanbegroupedintoavariablewithvaluesofonly0and1.Effectsandspecificgenerallinearmodels(withexamples):Effectsandspecificgeneralizedlinearmodels(withexamples):logistic(log-odds),log-hazard,andproportionalhazards(Cox)regressions(un)pairedttest;(multiplelinear)reg-ression;ANOVA;ANCOVAStatisticalmodelEffectofpredictorPredictorDependent"slope"(ratioperunitofpredictor)numericnominal

SelectedNYFitnessratiosofproportions,odds,rates,hazardsnominalnominalInjuredNYSex"slope"(differenceperunitofpredictor);correlationnumericcontinuousActivityAgedifferenceinmeansnominalcontinuousStrengthTrialPoissonregression"slope"(ratioperunitofpredictor)numericcount

MedalsCostratioofcountsnominalcountInjuriesSexThemostcommoneffectstatistic,fornumbers

withdecimals(continuousvariables).Differencewhencomparing

differentgroups,e.g.,patientsvshealthy.Changewhentrackingthesamesubjects.Differenceinthechangesincontrolledtrials.Thebetween-subjectstandarddeviation

providesdefaultthresholdsforimportant

differencesandchanges.Youthinkabouttheeffect(mean)intermsofa

fractionormultipleoftheSD(mean/SD).Theeffectissaidtobestandardized.Thesmallestimportanteffectis±0.20(±0.20ofanSD).TrialStrength

prepost1post2patientshealthyStrength(means&SD)differenceorchangeinmeansnominalcontinuousStrengthTrialEffectPredictorDependentExample:theeffectofatreatmentonstrength

strengthpostpreTrivialeffect(0.1xSD)

strengthpostpreVerylargeeffect(3.0xSD)Interpretationof

standardized

differenceor

changeinmeans:Cohen<0.2Hopkins<0.20.2-0.50.2-0.60.5-0.80.6-1.2>0.81.2-2.0?2.0-4.0trivialsmallmoderatelargeverylarge?>4.0extremelylarge2.04.0trivialsmallmoderatelargeverylargeext.largeCompletescale:Relationshipofstandardizedeffectto

differenceorchangeinpercentile:

strength

area

=50%athlete

on50thpercentile

strength

Standardized

effect

=0.20athlete

on58thpercentilearea

=58%0.2080850.2095971.0050842.005098Standardized

effect0.20Percentile

change50580.255060Can'tdefinesmallesteffectforpercentiles,becauseitdependswhatpercentileyouareon.Butit'sagoodpracticalmeasure.AndeasytogeneratewithExcel,

ifthedataareapprox.normal.CautionswithStandardizing

ChoiceoftheSDcanmakeabigdifferencetotheeffect.Usethebaseline(pre)SD,nevertheSDofchangescores.StandardizingworksonlywhentheSDcomesfromasamplerepresentativeofawell-definedpopulation.Theresultingmagnitudeappliesonlytothatpopulation.Bewareofauthorswhoshowstandarderrorsofthemean(SEM)ratherthanSD.SEM=SD/(samplesize)Soeffectslookalotbiggerthantheyreallyare.Checkthefineprint;ifauthorshaveshownSEM,dosomementalarithmetictogettherealeffect.OtherSmallestDifferencesorChangesinMeansSingle5-to7-ptLikertscales:halfastep.Visual-analogscalesscoredas0-10:1unit.Athleticperformance…MeasuresofAthleticPerformanceForfitnesstestsofteam-sport

athletes,usestandardization.Fortopsoloathletes,anenhancementthatresultsinoneextramedalper10competitionsisthesmallestimportanteffect.Simulationsshowthisenhancementisachievedwith0.3ofanathlete'stypicalvariabilityfromcompetitiontocompetition.Example:ifthevariabilityisacoefficientofvariationof1%,thesmallestimportantenhancementis0.3%.NotethatinmanypublicationsIhavemistakenlyreferredto0.5ofthevariabilityasthesmallesteffect.Moderate,large,verylargeandextremelylargeeffectsresultinanextra3,5,7and9medalsinevery10competitions.Thecorrespondingenhancementsasfactorsofthevariabilityare:2.54.0trivialsmallmoderatelargeverylargeext.largeBeware:smallesteffectonathleticperformancedependsonmethodofmeasurement,because…Apercentchangeinanathlete'sabilitytooutputpowerresultsindifferentpercentchangesinperformanceindifferenttests.Thesedifferencesareduetothepower-durationrelationshipforperformanceandthepower-speedrelationshipfordifferentmodesofexercise.Example:a1%changeinendurancepoweroutputproducesthefollowingchanges…1%inrunningtime-trialspeedortime;~0.4%inroad-cyclingtime-trialtime;0.3%inrowing-ergometertime-trialtime;~15%intimetoexhaustioninaconstant-powertest.Ahard-to-interpretchangeinanytestfollowingafatiguingpre-load.Aslopeismorepracticalthanacorrelation.Butunitofpredictorisarbitrary,soit's

hardtodefinesmallesteffectforaslope.Example:-2%peryearmayseemtrivial,

yet-20%perdecademayseemlarge.Forconsistencywithinterpretationofcorrelation,

bettertoexpressslopeasdifferencepertwoSDsofpredictor.Itgivesthedifferencebetweenatypicallylowandhighsubject.Seethepageoneffectmagnitudesatformore.Easiertointerpretthecorrelation,usingCohen'sscale.Smallestimportantcorrelationis±0.1.Completescale:Butnote:invaliditystudies,correlations>0.90aredesirable.EffectPredictorDependent"slope"(differenceperunitofpredictor);correlationnumericcontinuousActivityAgeAgeActivity

r

=

-0.50.70.9trivialsmallmoderatelargeverylargeext.largeTheeffectofanominalpredictorcanalsobeexpressedasacorrelation=√(fractionof“varianceexplained〞).A2-levelpredictorscoredas0and1givesthesamecorrelation.Withequalnumberofsubjectsineachgroup,thescalesforcorrelationandstandardizeddifferencematchup.For>2levels,thecorrelationcan’tbeappliedtoindividuals.Avoid.Correlationswhencontrollingforsomething…Interpretingslopesanddifferencesinmeansisnogreatproblemwhenyouhaveotherpredictorsinthemodel.BecarefulaboutwhichSDyouusetostandardize.Butcorrelationsareachallenge.Thecorrelationiseitherpartialorsemi-partial(SPSS:"part").Partial=effectofthepredictorwithinavirtualsubgroupofsubjectswhoallhavethesamevaluesoftheotherpredictors.Semi-partial=uniqueeffectofthepredictorwithallsubjects.Partialisprobablymoreappropriatefortheindividual.Confidencelimitsmaybeaprobleminsomestatspackages.TheNamesofLinearModelswithaContinuousDependentYouneedtoknowthejargonsoyoucanusetherightprocedureinaspreadsheetorstatspackage.Unpairedttest:for2levelsofasinglenominalpredictor.Usetheunequal-variancesversion,nevertheequal-variances.Pairedttest:asabove,butthe2levelsareforthesamesubjects.Simplelinearregression:asinglenumericpredictor.Multiplelinearregression:2ormorenumericpredictors.Analysisofvariance(ANOVA):oneormorenominalpredictors.Analysisofcovariance(ANCOVA):oneormorenominalandoneormorenumericpredictors.Repeated-measuresanalysisof(co)variance:AN(C)OVAinwhicheachsubjecthastwoormoremeasurements.Generallinearmodel(GLM):anycombinationofpredictors.InSPSS,nominalpredictorsarefactors,numericsarecovariates.Mixedlinearmodel:anycombinationofpredictorsanderrors.TheErrorTerminLinearModelswithaContinuousDependentStrength=a+b*Ageisn’tquiterightforrealdata,because

nosubject’sdatafitthisequationexactly.What’smissingisadifferenterrorforeachsubject:

Strength=a+b*Age+errorThiserrorisgivenanoverallmeanofzero,anditvariesrandomly(positiveandnegative)fromsubjecttosubject.It’scalledtheresidualerror,andthevaluesaretheresiduals.residual=(observedvalue)minus(predictedvalue)Inmanyanalysestheerrorisassumedtohavevaluesthatcomefromanormal(bell-shaped)distribution.Thisassumptioncanbeviolated.Testingfornormalityissilly.TheCentralLimitTheoremassuresanormalsamplingdistribution.Withacountasthedependent,theerrorhasaPoissondistribution,whichisanissueAddresswithgeneralizedlinearmodeling–seelater.Youcharacterizetheerrorwithastandarddeviation.It’salsoknownasthestandarderroroftheestimateortherootmeansquareerror.Ingenerallinearmodels,theerrorisassumedtobeuniform.Thatis,thereisonlyoneSDfortheresiduals,ortheerrorforeverydatumisdrawnfromasingle“hat〞.Non-uniformerrorisknownasheteroscedasticity.Ifyoudon’tdosomethingaboutit,yougetwronganswers.Withoutspecialtreatment,manydatasetsshowbiggererrorsforbiggervaluesofthedependent.ThisproblemisobviousinsometablesofmeansandSDs,inscatterplots,orinplotsofresidualvspredictedvalues(seelater).Suchplotsofindividualvaluesarealsogoodforspottingoutliers.Itarisesfromthefactthateffectsanderrorsinthedataarepercentsorfactors,notabsolutevalues.Example:anerrororeffectof5%is5sin100sbut10sin200s.Addresstheproblembyanalyzingthelog-transformeddependent.5%effectmeansPost=Pre*1.05.Thereforelog(Post)=log(Pre)+log(1.05).Thatis,theeffectisthesameforeveryone:log(1.05).Andwenowhavealinear(additive)model,notanon-linearmodel,sowecanuseallourusuallinearmodelingprocedures.A5%errormeanstypically1.05and1.05,or1.05.Anda100%errormeanstypically2.0(i.e.,valuesvarytypicallybyafactorof2),andsoon.Whenyoufinishanalyzingthelog-transformeddependent,youback-transformtoapercentorfactoreffectusingexponentiale.Showpercentsforanythingupto~30%.Showfactorsotherwise,e.g.,whenthedependentisahormoneconcentration.Usethelog-transformedvalueswhenstandardizing.Logtransformationisoftenappropriateforanumericpredictor.Theeffectofthepredictoristhenexpressedperpercent,per10%,per2-foldincrease,andsoon.Exampleofsimplelinearregressionwithadependentrequiringlogtransformation.Alogscaleorlogtransformationproducesuniformresiduals.020040060080010003456789Predictor100*ln(Dependent)1101001000100003456789PredictorDependent(logscale)-2000-10000100020003000020004000PredictedsResiduals-100-500501002505007501000PredictedsResiduals-200002000400060003456789PredictorDependentResidualPredictedNon-uniformscatterUniformscatterRanktransformationfornon-normalityandnon-uniformity?Sortallthevaluesofthedependentvariable,rankthem(i.e.,numberthem1,2,3,…),thenusethisrankinallfurtheranalyses.Theresultinganalysesaresometimescallednon-parametric.Butit’sstilllinearmodeling,soit’sreallyparametric.TheyhavenameslikeWilcoxonandKruskal-Wallis.Somearetrulynon-parametric:thesigntest;neural-netmodeling.Someresearchersthinkyouhavetousethisapproachwhen“thedataarenotnormallydistributed〞.Infact,therank-transformeddependentisanythingbutnormallydistributed:ithasauniform(flat)distribution!!!Doesranktransformationdealwithuniformityofeffectsanderror?No!Example:with100observations,thereisnowaythedifferencebetweenrank1and2isthesameeffectasthedifferencebetween50and51(or99and100,forathleticperformance).SoNEVERuserawranktransformation.Butlog(rank)appearstoworkwellforathleticperformance.Non-uniformityalsoariseswithdifferentgroupsandtimepoints.Example:asimplecomparisonofmeansofmalesandfemales,withdifferentSDformalesandfemales(evenafterlogtransformation).Hencetheunequal-varianceststatisticortest.Toincludecovariateshere,youcan’tusethegenerallinearmodel:

youhavetokeepthegroupsseparate,asinmyspreadsheets.Example:acontrolledtrial,withdifferenterrorsatdifferenttimepointsarisingfromindividualresponsesandchangeswithtime.MANOVAandrepeated-measuresANOVAcangivewronganswers.Addressbyreducingorcombiningrepeatedmeasurementsintoasinglechangescoreforeachsubject:within-subjectmodeling.ThenallowfordifferentSDofchangescoresbyanalyzingthegroupsseparately,asabove.Bonus:youcancalculateindividualresponsesasanSD.SeeRepeatedMeasuresandRandomEffectsatand/orthearticleonthecontrolled-trialspreadsheetsformore.Orspecifyseveralerrorsandmuchmorewithamixedmodel...Mixedmodelingisthecutting-edgeapproachtotheerrorterm.Mixed=fixedeffects+randomeffects.Fixedeffectsaretheusualtermsinthemodel;theyestimatemeans.Fixed,becausetheyhavethesamevalueforeveryoneinagrouporsubgroup;theyarenotsampledrandomly.Randomeffectsareerrortermsandanythingelserandomlychosenfromsomepopulation;eachissummarizedwithanSD.Thegenerallinearmodelallowsonlyoneerror.Mixedmodelsallow:specificationofdifferenterrorsbetweenandwithinsubjects;within-subjectcovariates(GLMallowsonlysubjectcharacteristicsorothercovariatesthatdonotchangebetweentrials);specificationofindividualresponsestotreatmentsandindividualdifferencesinsubjects’trends;interdependenceoferrorsandotherrandomeffects,whichariseswhenyoumodeldifferentlinesorcurvesforeachsubject.Withrepeatedmeasurementincontrolledtrials,simplifyanalysesbyanalyzingchangescores,evenwhenusingmixedmodeling.Example:adependentscoredas0or1(injurednooryes)predictedbysex(female,male)ofplayers

inaseasonoftouchrugby.Convertthe0sand1sineachgroup

toproportionsbyaveraging,thenmultiply

by100toexpressaspercents.RiskdifferenceorproportiondifferenceAcommonmeasure.Example:a

-

b=75%

-

36%=39%.Problem:thesenseofmagnitudeofagivendifferencedependsonhowbigtheproportionsare.Example:fora10%difference,90%vs80%doesn'tseembig,but… 11%vs1%canbeinterpretedasahuge"difference"(11xtherisk).Problem:nowaytomodelproportions(apartfromattest).differenceofproportions;ratiosofproportions,odds,rates,hazards,meaneventtimenominalnominalInjuredNYSexEffectPredictorDependentmalefemaleProportion

injured(%)Sex0100a=

75%b=

36%Riskratio(relativerisk)orproportionratioAnothercommonmeasure.

Example:a/b=75/36=2.1,whichmeans

malesare"2.1timesmorelikely"tobeinjured,

or"a110%increaseinrisk"ofinjuryformales.Problem:ifit'satimedependentmeasure,andyou

waitlongenough,everyonegetsaffected,soriskratio=1.00.Butitworksforraretime-dependentrisksandfortime-independentclassifications(e.g.,proportionplayingasport).Smallestimportanteffect:

forevery10injuredmalesthereare9injuredfe